Magnetic higher-order nodal lines
MMagnetic higher-order nodal lines
Zeying Zhang, Zhi-Ming Yu, ∗ and Shengyuan A. Yang College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing 100029, China Key Lab of Advanced Optoelectronic Quantum Architecture and Measurement (MOE),Beijing Key Lab of Nanophotonics & Ultrafine Optoelectronic Systems,and School of Physics, Beijing Institute of Technology, Beijing 100081, China Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore
Nodal lines, as one-dimensional band degeneracies in momentum space, usually feature a linearenergy splitting. Here, we propose the concept of magnetic higher-order nodal lines, which are nodallines with higher-order energy splitting and realized in magnetic systems with broken time reversalsymmetry. We provide sufficient symmetry conditions for stabilizing magnetic quadratic and cubicnodal lines, based on which concrete lattice models are constructed to demonstrate their existence.Unlike its counterpart in nonmagnetic systems, the magnetic quadratic nodal line can exist as theonly band degeneracy at the Fermi level. We show that these nodal lines can be accompaniedby torus surface states, which form a surface band that span over the whole surface Brillouinzone. Under symmetry breaking, these magnetic nodal lines can be transformed into a variety ofinteresting topological states, such as three-dimensional quantum anomalous Hall insulator, multiplelinear nodal lines, and magnetic triple-Weyl semimetal. The three-dimensional quantum anomalousHall insulator features a Hall conductivity σ xy quantized in unit of e / ( hd ) where d is the latticeconstant normal to the x - y plane. Our work reveals previously unknown topological states, andoffers guidance to search for them in realistic material systems. I. INTRODUCTION
Topological metals and semimetals have been attract-ing significant interest in current research [1–4]. Thesestates are characterized by the symmetry/topology pro-tected band degeneracies near the Fermi level, becausethey determine the low-energy quasiparticle excitationsand hence the physical properties of the system [5–10].Therefore, a central task in the field is to discover andclassify all possible types of protected band degeneracies.Such degeneracies can be classified from different per-spectives. For example, regarding the dimensionality ofthe degeneracy manifold in momentum space, the banddegeneracies can be classified into zero-dimensional (0D)nodal points [11–15], 1D nodal lines [16–22], or even 2Dnodal surfaces [23–27]. For each class, further classifica-tion can be made based on the number of degeneracy, theresulting Fermi surface topology, the distribution in theBrillouin zone (BZ), and etc.In the classification scheme, the character of band dis-persion clearly plays an important role, as it directly af-fects the density of states, the group velocity, and pos-sible topological charge of the low-energy quasiparticles.For most cases, the dispersion around a band degeneracyis of linear type, namely, the degeneracy is formed by thelinear crossing between two bands. Nevertheless, undercertain symmetries, the linear order term may be forbid-den, and then the leading order dispersion in the bandenergy splitting will be pushed to higher orders [28–32].For example, it was found that there exist symmetry-protected twofold nodal points, known as multi-Weyl ∗ zhiming [email protected] points, around which the leading order band splittingis quadratic or even cubic along certain directions [29].Similar study was later extended to fourfold Dirac points[32–34], and a systematic classification of higher-orderDirac points was achieved in Ref. [35].Recently, in Ref. [36], Yu et al. discovered the possibil-ity of higher-order nodal lines in nonmagnetic systems.They found that quadratic or cubic dispersion could bethe leading order dispersion of band splitting in the trans-verse plane for every point on the line. The special dis-persion leads to many interesting effects, including dis-tinct scalings in the thermodynamic and response prop-erties, unusual surface states, and rich topological phasesresulted from symmetry breaking [36, 37].The study of Ref. [36] is restricted to nonmagnetic sys-tems which preserve the time reversal symmetry T . Formagnetic systems, T is broken, which will fundamentallyimpact the topological classification. A trend in recentresearch is to push the study of topological phases tomagnetic systems [38–45]. Particularly, the magnetic lin-ear nodal lines have been actively explored in magneticmaterials [46–54].In view of these recent advances, a natural questionis: Is it possible to have protected magnetic higher-ordernodal lines?
In this work, we answer the above question in the af-firmative. We show that nodal lines with quadratic (cu-bic) leading order dispersion can be realized in magneticsystems with spin-orbit coupling fully considered, whichwe term as magnetic quadratic (cubic) nodal lines. Weprovide sufficient symmetry conditions for their protec-tion, based on which we construct lattice models to ex-plicitly demonstrate the existence of these nodal lines.Notably, unlike nonmagnetic systems, we find that mag-netic systems can host quadratic nodal lines as the only a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n band degeneracy at Fermi level. Both quadratic and cu-bic nodal lines may be accompanied by a special kind oftorus surface states, which span over the whole surfaceBZ and lead to a large surface density of states. Further-more, under symmetry breaking, these magnetic nodallines can evolve into a variety of intriguing topologicalstates, such as the 3D quantum anomalous Hall (QAH)insulator, multiple linear nodal lines, and the magnetictriple-Weyl semimetal. Particularly, the 3D QAH insu-lator features a Hall conductivity σ xy quantized in unitof e / ( hd ), with d the lattice constant normal to the x - y plane. Our work reveals previously unknown topologicalphases in systems with T breaking, offers detailed guid-ance to search for them in real material systems, and willstimulate further studies on their fascinating properties. II. MAGNETIC QUADRATIC NODAL LINEA. Symmetry condition
Let’s first present a symmetry condition that can pro-tect a magnetic quadratic nodal line (MQNL) in mag-netic systems. The symmetries should help to protectthe degeneracy along a 1D line and also eliminate the lin-ear term in the band splitting around this line. Here, wefind that a MQNL can be stabilized on a high-symmetryline that is an invariant subspace of two symmetries: athree fold rotation C z and a magnetic symmetry M z T involving a mirror, where we take the rotation axis to bealong the z direction. Here, M z T is a combined symme-try, while the individual M z and T symmetries are bro-ken by the magnetism. The two symmetries restrict theMQNL to the Γ- A path of the hexagonal BZ, as shownin Fig. 1(c).To explicitly demonstrate the quadratic dispersion, wetake an arbitrary point Q on the Γ- A path. The Blochstates at Q can be chosen as the eigenstates of C z . Formagnetic systems, it is necessary to consider the spin-orbit coupling. Then the C z eigenvalues are given by c z = − , e ± iπ/ , and we denote the eigenstates as | c z (cid:105) by using the eigenvalues. Since C z commutes with M z T ,we have C z ( M z T | c z (cid:105) ) = c ∗ z ( M z T | c z (cid:105) ) , (1)which indicates that the two states | e iπ/ (cid:105) and | e − iπ/ (cid:105) must always form a pair, degenerate in energy. Hence,this degeneracy leads to a doubly degenerate nodal linealong the Γ- A path. In the basis of these two degener-ate states, the matrix representations of the symmetryoperators are given by C z = e iσ π/ , M z T = σ K , (2)with σ i ’s the Pauli matrices and K the complex conju-gation operator. Then, the effective Hamiltonian con-strained by C z and M z T around Q in the transverseplane can be obtained as H Q eff ( k ) = ck (cid:107) + αk − σ + + α ∗ k σ − , (3) (a) )b( )c( a b a b c Γ 〜 X 〜 U 〜 Z 〜 M KL H A Γ Q
42 13
P-P
FIG. 1. (a) Side view and (b) top view of the unit cell for theMQNL model. The red arrows denote the magnetic momentson the sites. (c) Bulk and surface BZ of the lattice model. where k (cid:107) = (cid:113) k x + k y , c ( α ) is a real (complex) modelparameter which generally depends on Q , k ± = k x ± ik y ,and σ ± = ( σ ± iσ ) /
2. The spectrum of this Hamiltonian(3) is E = ( c ± | α | ) k (cid:107) , (4)which confirms that the leading order dispersion in theband splitting is quadratic and thus the nodal line hereis indeed a MQNL.Before proceeding, we note that although the presenceof the above MQNL does not require a vertical mirrorsymmetry (denoted as M x without loss of generality),MQNLs are compatible with this symmetry such as inthe lattice model to be discussed in a while. When we dohave this symmetry, its matrix representation in the basisof the degenerate pair will be given by M x = − iσ , andone can show that the form of the effective Hamiltonian(3) remains unchanged, but α will be constrained to bea real number. B. Lattice model for MQNL
We construct a lattice model with the symmetry con-dition specified above to explicitly demonstrate the exis-tence of the MQNL. Consider a 3D lattice consisting of2D honeycomb lattices AA-stacked along the z direction,as shown in Fig. 1(a-b). Here, each unit cell containstwo layers and four active sites. The four sites are la-beled as 1 to 4, as in Fig. 1(a). At each site, we put one s -like basis orbital φ s with a specific spin polarization,such that the basis of our model in a unit cell is given byΦ = ( φ s (1) | ↑(cid:105) , φ s (2) | ↓(cid:105) , φ s (3) | ↓(cid:105) , φ s (4) | ↑(cid:105) ). Physically,this case may correspond to a G-type antiferromagnetic(AFM) order as illustrated in Fig. 1(a).We require that the model respects the C z and M z T symmetries. In the basis of Φ, these symmetry operatorstake the form of C z = e i Γ π/ , M z T = i Γ K , (5)where we define Γ µν ≡ σ µ ⊗ σ ν and σ denotes the 2 × M x , whichinterchanges A and B sites in each layer, such that it isrepresented by M x = i Γ . (6)Following the standard approach as in Refs. [55, 56], we can construct the lattice model that satisfies the sym-metries in Eqs. (5) and (6). In momentum space, theobtained model can be written as H = A − A sin k y (cid:18) sin k x √ + cos k x √ (cid:19) + A √ (cid:20)(cid:18) sin k x √ k x √ k y (cid:19) Γ − (cid:18) cos k x √ − cos k x √ k y (cid:19) Γ (cid:21) + (cid:18) cos k x √ k x √ k y (cid:19) (cid:18) A cos k z + A sin k z (cid:19) − (cid:18) sin k x √ − k x √ k y (cid:19) (cid:18) A cos k z + A sin k z (cid:19) . (7)Here, A i ( i = 0 , , ,
3) are real model parameters. Onecan readily check that the Hamiltonian (7) is invariantunder the symmetry operators in Eqs. (5) and (6).The calculated band structure of this model is plottedin Fig. 2(a). One observes that both lower two bands andupper two bands form twofold nodal lines along the Γ- A path. One can check that these nodal lines are indeedMQNLs. In Fig. 2(b), we plot the dispersion around ageneric point Q [marked in Fig. 2(a)] on the lower nodalline, which confirms the quadratic band splitting.In fact, one can expand the model (7) around point Q to obtain a two-band effective model in the plane trans-verse to the nodal line. As shown in Appendix A, theresult recovers the effective model in Eq. (3), confirmingthat this is indeed the desired MQNL.It should be pointed out that if we focus on the lowertwo bands in Fig. 2(a), the MQNL on Γ- A is their only de-generacy in the BZ. When the electron filling has only thelowest band filled, i.e., the Fermi level is around the nodalline, then the quasiparticles around the MQNL will playa dominant role in the physical properties of the system.In comparison, in nonmagnetic systems, the two bandsthat form a quadratic nodal line must also degenerate onother paths (such as the nodal line lying in Γ M K plane) (a) (b) E n e r g y L M Γ A H K E n e r g y P Q -P-0.0100.01 l o g ( Δ E ) log(q)ΔE~q Q FIG. 2. (a) Band structure of the lattice model (7) alonghigh-symmetry lines. (b) Enlarged view of the band structurealong P - Q path [marked in Fig. 1(c)] which lies in the planeperpendicular to k z . The inset shows the log-log plot forthe band splitting ∆ E versus the momentum deviation fromthe nodal line. In the model, we set A = 0 . A = 0 . A = 0 .
1, and A = 0 . in the BZ, due to the presence of M z [36]. In other words,the quadratic nodal line in nonmagnetic systems must beaccompanied by other degeneracies in the BZ, whereas itcan be the only one for magnetic systems. III. MAGNETIC CUBIC NODAL LINEA. Symmetry condition
Next, we investigate the possibility of cubic nodal linesin magnetic systems. In the following, we show that mag-netic cubic nodal lines (MCNLs) can be protected in theinvariant subspace of two symmetries: a sixfold rotation C z and a vertical mirror M x that contains the rotationaxis. Hence, in the BZ, such MCNL can only appear onthe Γ- A path. Note that these symmetries do not involvethe time reversal, so the symmetry condition here actu-ally applies for both nonmagnetic and magnetic systems.Let’s consider a generic point Q on the high-symmetrypath Γ- A , which has both C z and M x symmetries. TheBloch states at Q can be chosen as the eigenstates of C z . We may denote these eigenstates | c z (cid:105) by the C z eigenvalues, with c z = ± i, e ± iπ/ , − e ± iπ/ . We havethe following relations between C z and M x , C z M x = M x C − z , (8) (a) )b( )c( a b a b c Γ 〜 X 〜 U 〜 Z 〜 M KL HΓ Q P-P A FIG. 3. (a) Side view and (b) top view of the unit cell for theMCNL model. The directions of local moments are denotedby the red arrows. (c) Bulk and surface BZ of the latticemodel. so that C z ( M x | c z (cid:105) ) = M x (cid:0) C − z | c z (cid:105) (cid:1) = c ∗ z ( M x | c z (cid:105) ) . (9)Hence, each state | c z (cid:105) at Q must have a degenerate part-ner with eigenvalue c ∗ z , which leads to twofold nodal lineson the Γ- A path. And all these | c z (cid:105) ’s form three degen-erate pairs: ( | i (cid:105) , | − i (cid:105) ), ( | ε (cid:105) , | ε ∗ (cid:105) ), and ( | − ε (cid:105) , | − ε ∗ (cid:105) ),where ε = e iπ/ .The MCNL corresponds to the degenerate pair witheigenvalues c z = ± i . In the two-state basis ( | i (cid:105) , | − i (cid:105) ),the matrix representation of the symmetry operators canbe written as C z = iσ , M x = iσ . (10)Then the effective Hamiltonian at Q constrained by thesesymmetries in the plane transverse to the nodal line isobtained as H Q eff ( k ) = c k (cid:107) + [ i ( c k − + c k ) σ + + h.c. ] , (11)where c i ’s ( i = 1 , ,
3) are real model parameter. Notethat the first (quadratic) term is proportional to the iden-tity matrix, so it does not affect the leading order of theband splitting. Indeed, the spectrum of (11) is E = c k (cid:107) ± | ( c k − + c k ) | , (12)showing that leading order in the band splitting is cubic.Thus, this nodal line on the Γ- A path is a MCNL. B. Lattice model for MCNL
Guided by the above symmetry condition, we constructa concrete lattice model to demonstrate the existenceof the MCNL. Consider the 3D lattice as illustrated inFig. 3(a-b), formed by stacking 2D triangular latticesalong the z direction. Let’s consider the A-type AFMordering as in Fig. 3(a). This preserves the C z symme-try and also a glide mirror symmetry ˜ M x = { M x | } .Note that the glide character of ˜ M x is not essential forthe MCNL. Indeed, the symmetry analysis in the previ-ous section is not affected by the fractional translation ofthe glide mirror.In this lattice model, a unit cell contains two sites,labeled as 1 and 2 as in Fig. 3(a). We put one basisorbital on each site: at site 1, it is the p + orbital withspin up; and at site 2, it is the p − orbital with spindown. In other words, the basis of our model is takento be Φ = ( φ p + (1) | ↑(cid:105) , φ p − (2) | ↓(cid:105) ). This setup conformswith the specified symmetries above. In this basis, thesymmetry operators take the form of C z = iσ , ˜ M x = iσ , (13)which are the same as in Eq. (10). Constrained by thesesymmetries, the obtained lattice model in momentumspace can be expressed as H = A + A cos k z + A (2 cos √ k x k y k y ) + A cos k z (cid:32) sin k y − √ k x k y (cid:33) σ + A cos k z (cid:34) √ k x k y − sin( √ k x ) (cid:35) σ . (14)Again, the A i ’s here are real model parameters. Onecan readily check that the Hamiltonian (14) is invariantunder the symmetry operators in Eq. (13).The calculated band structure of this model (14) isshown in Fig 4(a). One can clearly observe the nodal lineon the Γ- A path. By checking the dispersion around theline, one can verify the leading order band splitting is ofcubic order [see Fig. 4(b)]. The cubic character can alsobe confirmed analytically by expanding the model (14)around a generic point Q on the Γ- A path. The obtainedeffective model exactly recovers that in Eq. (11). Here,if we assume the electron filling is one electron per unitcell, i.e., half-filling of the bands, then this MCNL willbe staying around the Fermi level.Besides the MCNL on Γ- A , in Fig. 4(a), one observesthere are additional degeneracies between the two bandson the L - M and A - H paths. The degeneracies on L - M correspond to an essential nodal line. This path has C v symmetry, which only has a two-dimensional irreducibledouble representation Γ [57], so all bands here must ac-quire a twofold degeneracy. The nodal line on L - M is aconventional linear nodal line. Hence, unlike the MQNLdiscussed in Sec. II, the MCNL must coexist with addi-tional linear nodal lines in the band structure.As for the degeneracies on A - H , we find that it actu-ally correspond to a nodal surface on the k z = π plane.This magnetic nodal surface is protected by the C z T (cid:48) symmetry, Such kind of magnetic nodal surface was firstproposed by Wu et al. in Ref [25]. Here, we note that T (cid:48) is an extra symmetry of the lattice model, which isactually not required for the MCNL. Thus, the nodal sur-face can be removed by breaking the T (cid:48) symmetry whilemaintaining C z and ˜ M x and hence the existence of theMCNL. E n e r g y L M Γ A H K E n e r g y P Q -P-0.020.0.02 -0.600.6 l o g ( Δ E ) log(q)ΔE~q (a) (b) Q FIG. 4. (a) Band structure of the lattice model (14) alonghigh-symmetry lines. (b) Enlarged view of the band structurealong the P - Q path [marked in Fig. 3(c)] which is in the planeperpendicular to k z . The inset shows the log-log plot for theband splitting ∆ E versus the momentum deviation from thenodal line. In the model, we take A = 0 . A = 0 . A = − . A = 0 .
3, and A = 0 . UΓ X Z Γ U~ ~ ~ ~ ~ ~ -0.150.150 Γ 〜 X 〜 U 〜 Z 〜 -π π E n e r g y Z a k p h a s e (a) (b)(c) (d) ΓA -π0π -X~ X~ Γ~ π2 FIG. 5. (a) Schematic figure showing the MQNL in the BZ.(b) Zak phase Z ( k y , k z ) as a function of k y with k z = 0. (c)Projected spectrum on the (1010) surface. The red arrowsindicate the torus surface states, which span over the wholesurface BZ, as illustrated in (d). In (d), we also indicate thecalculated Zak phase for lines perpendicular to the (1010)surface. IV. TORUS SURFACE STATES
Materials with conventional linear nodal lines oftenhost drumhead type surface states [17, 18]. Here, “drum-head” means these states occupy a finite region in thesurface BZ bounded by the projection of the nodal lineon that surface. The stability of these drumhead sur-face states is often enforced by the nontrivial Zak phase,which is the Berry phase along a straight line crossingthe bulk BZ and perpendicular to the specified surface.For example, to study the (1010) surface normal to the x direction, one may examine the Zak phase [58] Z ( k y , k z ) = (cid:88) n ∈ occ (cid:73) (cid:104) u n ( k ) | i∂ k x | u n ( k ) (cid:105) dk x , (15)where | u (cid:105) is the cell-periodic Bloch state, the integrationis over the line with fixed k y and k z , and the summation Z a k p h a s e -π0π -X~ X~Γ~Z ~~ -0.40.40 Γ~ E n e r g y (a) (b) Γ ~ X
FIG. 6. (a) Projected spectrum on the (1010) surface forthe MCNL model. The red arrows indicate the torus surfacestates. (b) Zak phase Z ( k y , k z ) as a function of k y with k z =0. is over the occupied bands.The Zak phase is quantized in units of π under certainsymmetries. This is the case when the system has bothspin rotational symmetry and spacetime inversion sym-metry. Another case which is relevant to our discussionhere is when the system has a mirror plane normal to thestraight line on which the Zak phase is defined. For thesecases, an obtained nontrivial Zak phase Z ( k y , k z ) = π would indicate that there is a surface state at ( k y , k z ) inthe BZ for the (1010) surface. And as a linear nodal linesfeatures a π Berry phase, it separates regions with Z = π and Z = 0 in the surface BZ, and the drumhead surfacestates reside in the region of Z = π .As shown in Ref. [36], the order of the nodal line di-rectly affects its Berry phase. Distinct from the linearnodal lines, the MQNL features a 2 π (equivalent to 0when mod 2 π ) Berry phase for a small loop surroundingit. Consequently, the possible surface states will also ex-hibit distinct features. In Fig. 5(c), we plot the spectrumfor the lattice model in Eq. (7) on the surface normal to x , in which one can clearly observe the surface band.By scanning the surface BZ, we find that this surfaceband covers the whole BZ. Since the surface BZ forms atorus T , these states may be termed as the torus surfacestates. The presence of torus surface states is consistentwith the calculated Zak phase as shown in Fig. 5(b) and5(d). Here, the red dashed line in Fig. 5(d) indicates theprojection of the MQNL in the surface BZ. The impor-tant point is that unlike the linear nodal line, the MQNLdoes not impose a π discontinuity in the Zak phase, there-fore Z can take the nontrivial value π in the whole surfaceBZ, leading to the torus surface states.Similar analysis can be performed for the lattice model(14) containing the MCNL. MCNL carries a π Berryphase, so one may expect to see drumhead surface statessimilar to that of linear nodal lines. In Fig. 6(a), we showthe result of the side surface normal to x . Interestingly,again, one finds that there exist torus surface states, andthe Zak phase is nontrivial in the whole surface BZ. Thereason is that as shown in Sec. III B, besides the MCNL,the system must also have essential linear nodal linesalong the L - M path. On the surface in Fig. 3(c), theMCNL and the linear nodal line project to the same line (cid:101) Γ- (cid:101) Z in the surface BZ. Hence, the discontinuity in theZak phase across (cid:101) Γ- (cid:101) Z becomes π + π = 0 mod 2 π [see L M Γ A H K -0.300.60.3
Γ(1,0)Γ(0,0) X~ ~ ~ -0.150.150 E n e r g y UΓ X Z Γ U ~ ~ ~ ~ ~ ~ -0.150.150 E n e r g y E n e r g y E n e r g y δ=-0.0008δ=-0.08δ=-0.008 Z a k p h a s e π -X Γ X -0.3 Γ-X X (a) (b) (c)(d) (e) (f ) -X~ Γ~ -X~ Γ 〜 X 〜 U 〜 Z 〜 π π0 FIG. 7. MQNL under symmetry breaking. (a-c) Breaking M z T transforms the MQNL into a 3D QAH insulator. (a) Bandstructure of the MQNL model (7) with perturbation (16). (b) shows the projected spectrum on the (1010) surface. In thecalculation, we set δ = − .
08. (c) The Wilson loop in the k z = 0 plane for different perturbation strength δ , showing a unitChern number. (d-f) Breaking C z transforms the MQNL into two linear nodal lines. (d) Band structure [along the pathmarked in (f)] and (e) the projected spectrum on the (1010) surface of MQNL model (7) with perturbation (18). (f) Schematicfigure showing the two linear nodal lines (green lines) and the corresponding surface state distribution the surface BZ (greencolored region). Fig. 6(b)], similar to that in Fig. 5(d). Therefore, torussurface states can appear in this MCNL system as well.We have shown that torus surface states are compati-ble with both MQNLs and MCNLs, and their existenceis indeed demonstrated in the two lattice models. How-ever, we must point out that MQNLs and MCNLs cannotguarantee the existence of torus surface states. It is pos-sible to have the Zak phase tuned to zero for the wholesurface BZ, then there will be no stable surface states forthe system.
V. TOPOLOGICAL PHASE TRANSITION
The magnetic higher-order nodal lines discovered hereare protected by multiple symmetries. Under symmetrybreaking, they may transform to other interesting bandfeatures. Particularly, the lack of time reversal symmetrymay generate new physics not possible in their nonmag-netic counterparts.Let’s first consider the MQNL. When we break the M z T symmetry, the degeneracy of the MQNL will belifted, and the system may be turned into an insula-tor. For example, in the model Eq. (7), we may breakthe M z T symmetry by adding the following perturbationterm: H = δ Γ , (16)where δ denotes the perturbation strength. The result-ing band structure is plotted in Fig. 7(a). One observesthat the original MQNL is fully gapped out and the sys-tem becomes an insulator. Interestingly, we find thatthis insulator is topologically nontrivial. In Fig. 7(c), we evaluate the Chern number C of the k z = 0 plane by us-ing the Wilson loop method. (Here, we have only thelowest band occupied.) The result shows that C = 1 isnontrivial. Correspondingly, there must exist chiral edgestate for this 2D subsystem, as verified in Fig. 7(b). Now,since the system has a global band gap, every 2D slice ofthe bulk BZ with a fixed k z must have the same Chernnumber C = 1. Thus, the side surface of the systemmust be covered by chiral boundary states propagatingin the same direction. This leads to a novel 3D QAHinsulator state, which features a vanishing longitudinalconductivity σ xx = 0 and an quantized anomalous Hallconductivity given by σ xy = (cid:90) π/d − π/d dk z π (cid:18) C e h (cid:19) = C e hd , (17)where we have explicitly written out the factor of 1 /d in the wave vector, with d the lattice constant along z .Thus, the quantization of σ xy for the 3D QAH state is inunits of e / ( hd ).We note that this 3D QAH state is analogous tothe 3D quantum Hall effect (QHE) originally proposedby Halperin [59] and recently demonstrated by Tang etal. [60]. Meanwhile, there are also important differences.The 3D QHE is realized under strong magnetic field withLandau band formation, whereas the 3D QAH state ex-ists without external magnetic field. Related to thispoint, for the 3D QHE, the quantization unit (and hence σ xy ) typically changes with the magnetic field strength,since the Landau band degeneracy scales linearly with B [59]. In comparison, for a 3D QAH insulator, the quan-tization unit is a fixed value, determined by the structureof the system. K~Γ~ M~
L M Γ A H K -0.30.0.3 (a) (b)(c)
Without perturbation With perturbation
C=3
A LM MΓ Γ
3π π NS C=-1 E n e r g y FIG. 8. MCNL semimetal evolves into a triple-Weylsemimetal under symmetry breaking. (a) Band structureof MCNL model (14) with perturbation (19). (b) showsthe projected spectrum on the (0001) surface, showing threeFermi arcs connecting the projections of the triple Weyl pointand three single Weyl points. In the calculation, we set δ = δ = 0 .
05. (c) Schematic figure showing the transitionfrom the MCNL semimetal to the triple-Weyl semimetal.
Another interesting case is to break the C z symmetryin model, while keeping the M x symmetry. A possibleperturbation term may be written as H = δ (cid:18) cos k z + sin k z (cid:19) . (18)With this term, the MQNL will split into two linearnodal lines lying in the M x mirror plane, as illustratedin Fig. 7(d). On the surface that is parallel to the mirrorplane, the original torus surface states will transform intodrumhead surface states [see Fig. 7(e,f)]. Compared toFig. 5(a), the splitting of MQNL opens a region betweenthe two linear nodal lines which has a trivial Zak phase.As a result, surface states now exist only in the regionoutside of the two nodal lines in the surface BZ.As for the MCNL in model, let us consider the casewhen both C z and ˜ M x symmetries are broken but C z and T (cid:48) symmetries are maintained. This allows the fol-lowing perturbation term H = δ sin k z σ − δ sin k y (cid:32) cos √ k x − cos k y (cid:33) σ . (19)In this case, the MCNL, the essential linear nodal lines,and the magnetic nodal surface are all destroyed. TheMCNL and the linear nodal lines respectively evolve intoa triple-Weyl point with Chern number C = 3 and threeconventional Weyl points with C = − k z = 0 plane, namely, the Γ and the three M points, which is due to the Kramers-like degeneracyfor T (cid:48) = − k z = 0 plane. In comparison, T (cid:48) = 1 in the k z = π plane, so there is no such degeneracy there.We note that this configuration with a single protectedtriple-Weyl point in the BZ is unique for magnetic sys-tems; for nonmagnetic system, the protected triple Weylpoint would appear at least in a pair, or comes with anodal surface [36]. VI. DISCUSSION AND CONCLUSION
In this work, we have theoretically demonstrated theexistence of symmetry protected MQNLs and MCNLs. Itremains an important task to identify realistic materialsthat can host these higher-order nodal lines. The sym-metry conditions found here will offer useful guidance forsuch material search. Besides the nodal lines, it is alsointeresting to search for the topological states identifiedin Sec. V, such as the 3D QAH state and the magnetictriple-Weyl point.In experiment, the order of dispersion and the sur-face states can be directly probed by the angle-resolvedphotoemission spectroscopy (ARPES) method [61]. Asdiscussed in Ref. [36], the higher-order nodal lines cangive different scalings in the joint density of states (DOS)and Landau level energies, which can be probed with in-frared optical spectroscopy. The torus surface states canlead to a large surface DOS, which can be probed by thescanning tunneling spectroscopy (STS) [62]. Such largesurface DOS could be beneficial for realizing surface mag-netism and surface high-temperature superconductivity.In conclusion, with symmetry analysis, we have pro-posed a new class of nodal lines, namely, the MQNLs andMCNLs in magnetic systems. We present sufficient sym-metry conditions for stabilizing these high-order nodallines, and construct concrete lattice models to furtherdemonstrate our proposals. Remarkably, we find theMQNL can exist as the only band degeneracy at theFermi level. Both MQNL and MCNL can give rise tonovel torus surface state, covering the whole surface BZ.Furthermore, these nodal lines may be regarded as par-ent phases for many other interesting topological phases.Under symmetry breaking, they can generate phases with3D QAH insulator state, multiple magnetic linear nodallines, and magnetic triple-Weyl points. Our work extendsthe scope of higher-order nodal lines to magnetic systems,reveals a route towards 3D QAH insulator state, and pro-vides useful guidance to search for magnetic topologicalmaterials.
ACKNOWLEDGMENTS
The authors thank K. Fredericks and D. L. Dengfor valuable discussions. This work is supported byChina Postdoctoral Science Foundation (Grant No.2020M670106), the NSF of China (Grant Nos. 12004028,12004035), Fundamental Research Funds for the CentralUniversities (ZY2018), Beijing Institute of TechnologyResearch Fund Program for Young Scholars, and Sin-gapore Ministry of Education AcRF Tier 2 (Grant No.MOE2019-T2-1-001).
Appendix A: Two-band effective Model of MQNL
Since the magnetic nodal line formed by the lowest twobands in Fig. 2(a) are well separated from the other bandsin energy, we can establish a two-band effective model tocapture the physics of the line. First, we expand thelattice model (7) around a point on the Γ- A path (up toquadratic order) H Γ A = A + A k x Γ , − k y Γ )+ A √ (cid:2)(cid:0) k x − k y (cid:1) Γ − k x k y Γ (cid:3) +3 A cos k z + 3 A sin k z , − k (cid:107) ( A k z + A k z ) , (A1) with k (cid:107) = (cid:113) k x + k y . From Eq. (A1), the eigenstates ofthe lowest two bands are found to beΨ = N − ( γ, , , T , Ψ = N − (0 , γ, , T , (A2)where γ = − (cid:112) A + ( A − A ) cos k z + A √ (cid:0) A cos k z + iA sin k z (cid:1) , (A3)and N is a normalization coefficient. Then the two-bandeffective model can be obtained as H eff ( k ) = (cid:20) (cid:104) Ψ |H Γ A ( k ) | Ψ (cid:105) (cid:104) Ψ |H Γ A ( k ) | Ψ (cid:105)(cid:104) Ψ |H Γ A ( k ) | Ψ (cid:105) (cid:104) Ψ |H Γ A ( k ) | Ψ (cid:105) (cid:21) = ck (cid:107) + A ( k σ − + k − σ + ) , (A4)which recovers the effective Hamiltonian of MQNL in theEq. (3). In this effective model, the parameters are foundto be c = (cid:112) A + ( A − A ) cos k z + A / (2 √ |N | ) and A = A / (4 √ |N | ). [1] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu,Classification of topological quantum matter with sym-metries, Rev. Mod. Phys. , 035005 (2016).[2] N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyland Dirac semimetals in three-dimensional solids, Rev.Mod. Phys. , 015001 (2018).[3] S. A. Yang, Dirac and Weyl Materials: FundamentalAspects and Some Spintronics Applications, SPIN ,1640003 (2016).[4] A. Burkov, Topological semimetals, Nat. Mater. , 1145(2016).[5] H. B. Nielsen and M. Ninomiya, The adler-bell-jackiwanomaly and weyl fermions in a crystal, Phys. Lett. B , 389 (1983).[6] G. E. Volovik, The Universe in a Helium Droplet (Clarendon Press, Oxford, 2003).[7] P. Hosur, S. A. Parameswaran, and A. Vishwanath,Charge transport in weyl semimetals, Phys. Rev. Lett. , 046602 (2012).[8] D. Son and B. Spivak, Chiral anomaly and classical nega-tive magnetoresistance of Weyl metals, Phys. Rev. B ,104412 (2013).[9] F. de Juan, A. G. Grushin, T. Morimoto, and J. E.Moore, Quantized circular photogalvanic effect in Weylsemimetals, Nat. Commun. , 15995 (2017).[10] Y. Liu, Z.-M. Yu, C. Xiao, and S. A. Yang, Quantized cir-culation of anomalous shift in interface reflection, Phys.Rev. Lett. , 076801 (2020).[11] X. Wan, A. M. Turner, A. Vishwanath, and S. Y.Savrasov, Topological semimetal and Fermi-arc surfacestates in the electronic structure of pyrochlore iridates,Phys. Rev. B , 205101 (2011).[12] S. M. Young, S. Zaheer, J. C. Teo, C. L. Kane, E. J. Mele,and A. M. Rappe, Dirac semimetal in three dimensions,Phys. Rev. Lett. , 140405 (2012). [13] Z. Wang, Y. Sun, X.-Q. Chen, C. Franchini, G. Xu,H. Weng, X. Dai, and Z. Fang, Dirac semimetal and topo-logical phase transitions in A Bi (A=Na, K, Rb), Phys.Rev. B , 195320 (2012).[14] Y. X. Zhao and Z. D. Wang, Topological classificationand stability of fermi surfaces, Phys. Rev. Lett. ,240404 (2013).[15] B. Bradlyn, J. Cano, Z. Wang, M. G. Vergniory,C. Felser, R. J. Cava, and B. A. Bernevig, Beyond Diracand Weyl fermions: Unconventional quasiparticles inconventional crystals, Science , aaf5037 (2016).[16] A. A. Burkov, M. D. Hook, and L. Balents, Topologicalnodal semimetals, Phys. Rev. B , 235126 (2011).[17] S. A. Yang, H. Pan, and F. Zhang, Dirac and Weyl Super-conductors in Three Dimensions, Phys. Rev. Lett. ,046401 (2014).[18] H. Weng, Y. Liang, Q. Xu, R. Yu, Z. Fang, X. Dai, andY. Kawazoe, Topological node-line semimetal in three-dimensional graphene networks, Phys. Rev. B , 045108(2015).[19] Y. Chen, Y. Xie, S. A. Yang, H. Pan, F. Zhang, M. L.Cohen, and S. Zhang, Nanostructured carbon allotropeswith weyl-like loops and points, Nano Lett. , 6974(2015).[20] R. Yu, H. Weng, Z. Fang, X. Dai, and X. Hu, Topologicalnode-line semimetal and dirac semimetal state in antiper-ovskite cu PdN, Phys. Rev. Lett. , 036807 (2015).[21] Y. Kim, B. J. Wieder, C. L. Kane, and A. M. Rappe,Dirac line nodes in inversion-symmetric crystals, Physicalreview letters , 036806 (2015).[22] K. Mullen, B. Uchoa, and D. T. Glatzhofer, Line of diracnodes in hyperhoneycomb lattices, Phys. Rev. Lett. ,026403 (2015).[23] Q.-F. Liang, J. Zhou, R. Yu, Z. Wang, and H. Weng,Node-surface and node-line fermions from nonsymmor-phic lattice symmetries, Phys. Rev. B , 085427 (2016). [24] C. Zhong, Y. Chen, Y. Xie, S. A. Yang, M. L. Co-hen, and S. B. Zhang, Towards three-dimensional Weyl-surface semimetals in graphene networks, Nanoscale ,7232 (2016).[25] W. Wu, Y. Liu, S. Li, C. Zhong, Z.-M. Yu, X.-L. Sheng,Y. X. Zhao, and S. A. Yang, Nodal surface semimetals:Theory and material realization, Phys. Rev. B , 115125(2018).[26] O. T¨urker and S. Moroz, Weyl nodal surfaces, Phys. Rev.B , 075120 (2018).[27] X. Zhang, Z.-M. Yu, Z. Zhu, W. Wu, S.-S. Wang, X.-L.Sheng, and S. A. Yang, Nodal loop and nodal surfacestates in the Ti Al family of materials, Phys. Rev. B ,235150 (2018).[28] G. Xu, H. Weng, Z. Wang, X. Dai, and Z. Fang, Chernsemimetal and the quantized anomalous Hall effect inHgCr Se , Phys. Rev. Lett. , 186806 (2011).[29] C. Fang, M. J. Gilbert, X. Dai, and B. A. Bernevig,Multi-Weyl Topological Semimetals Stabilized by PointGroup Symmetry, Phys. Rev. Lett. , 266802 (2012).[30] Q. Liu and A. Zunger, Predicted Realization of Cu-bic Dirac Fermion in Quasi-One-Dimensional Transition-Metal Monochalcogenides, Phys. Rev. X , 021019(2017).[31] H. He, C. Qiu, X. Cai, M. Xiao, M. Ke, F. Zhang, andZ. Liu, Observation of quadratic Weyl points and double-helicoid arcs, Nat. Commun. , 1820 (2020).[32] B.-J. Yang and N. Nagaosa, Classification of stable three-dimensional Dirac semimetals with nontrivial topology,Nat. Commun. , 5898 (2014).[33] Z. Gao, M. Hua, H. Zhang, and X. Zhang, Classificationof stable Dirac and Weyl semimetals with reflection androtational symmetry, Phys. Rev. B , 205109 (2016).[34] W. C. Yu, X. Zhou, F.-C. Chuang, S. A. Yang, H. Lin,and A. Bansil, Nonsymmorphic cubic Dirac point andcrossed nodal rings across the ferroelectric phase transi-tion in LiOsO , Phys. Rev. Mater. , 051201 (2018).[35] W. Wu, Z.-M. Yu, X. Zhou, Y. X. Zhao, and S. A. Yang,Higher-order Dirac fermions in three dimensions, Phys.Rev. B , 205134 (2020).[36] Z.-M. Yu, W. Wu, X.-L. Sheng, Y. X. Zhao, and S. A.Yang, Quadratic and cubic nodal lines stabilized by crys-talline symmetry, Phys. Rev. B , 121106(R) (2019).[37] J.-R. Wang, W. Li, and C.-J. Zhang, Possible instabilitiesin quadratic and cubic nodal-line fermion systems withcorrelated interactions, Phys. Rev. B , 085132 (2020).[38] N. Morali, R. Batabyal, P. K. Nag, E. Liu, Q. Xu,Y. Sun, B. Yan, C. Felser, N. Avraham, and H. Bei-denkopf, Fermi-arc diversity on surface terminations ofthe magnetic Weyl semimetal Co Sn S , Science ,1286 (2019).[39] J.-Y. You, C. Chen, Z. Zhang, X.-L. Sheng, S. A. Yang,and G. Su, Two-dimensional weyl half-semimetal andtunable quantum anomalous hall effect, Phys. Rev. B , 064408 (2019).[40] D. Liu, A. Liang, E. Liu, Q. Xu, Y. Li, C. Chen, D. Pei,W. Shi, S. Mo, P. Dudin, et al. , Magnetic weyl semimetalphase in a kagom´e crystal, Science , 1282 (2019).[41] I. Belopolski, K. Manna, D. S. Sanchez, G. Chang,B. Ernst, J. Yin, S. S. Zhang, T. Cochran, N. Shumiya,H. Zheng, et al. , Discovery of topological Weyl fermionlines and drumhead surface states in a room temperaturemagnet, Science , 1278 (2019). [42] Y. Xu, L. Elcoro, Z.-D. Song, B. J. Wieder, M. Vergniory,N. Regnault, Y. Chen, C. Felser, and B. A. Bernevig,High-throughput calculations of magnetic topologicalmaterials, Nature , 702 (2020).[43] L. Jin, X. Zhang, Y. Liu, X. Dai, L. Wang, andG. Liu, Fully spin-polarized double-weyl fermions withtype-III dispersion in the quasi-one-dimensional materi-als X RhF (X=K, Rb, Cs), Phys. Rev. B , 195104(2020).[44] P. Puphal, V. Pomjakushin, N. Kanazawa, V. Uk-leev, D. J. Gawryluk, J. Ma, M. Naamneh, N. C.Plumb, L. Keller, R. Cubitt, E. Pomjakushina, and J. S.White, Topological magnetic phase in the candidate weylsemimetal cealge, Phys. Rev. Lett. , 017202 (2020).[45] J. Zou, Z. He, and G. Xu, The study of magnetic topolog-ical semimetals by first principles calculations, npj Com-putational Materials , 96 (2019).[46] J. Wang, Antiferromagnetic topological nodal linesemimetals, Phys. Rev. B , 081107 (2017).[47] B. Wang, H. Gao, Q. Lu, W. Xie, Y. Ge, Y.-H. Zhao,K. Zhang, and Y. Liu, Type-i and type-ii nodal lines co-existence in the antiferromagnetic monolayer cras , Phys.Rev. B , 115164 (2018).[48] S.-S. Wang, Z.-M. Yu, Y. Liu, Y. Jiao, S. Guan, X.-L.Sheng, and S. A. Yang, Two-dimensional nodal-loop half-metal in monolayer mnn, Phys. Rev. Materials , 084201(2019).[49] C. Chen, Z.-M. Yu, S. Li, Z. Chen, X.-L. Sheng, and S. A.Yang, Weyl-loop half-metal in li (FeO ) , Phys. Rev. B , 075131 (2019).[50] B. Feng, R.-W. Zhang, Y. Feng, B. Fu, S. Wu,K. Miyamoto, S. He, L. Chen, K. Wu, K. Shimada,T. Okuda, and Y. Yao, Discovery of weyl nodal lines ina single-layer ferromagnet, Phys. Rev. Lett. , 116401(2019).[51] S. Nie, Y. Sun, F. B. Prinz, Z. Wang, H. Weng, Z. Fang,and X. Dai, Magnetic Semimetals and Quantized Anoma-lous Hall Effect in EuB , Phys. Rev. Lett. , 076403(2020).[52] T. He, X. Zhang, Y. Liu, X. Dai, G. Liu, Z.-M. Yu, andY. Yao, Ferromagnetic hybrid nodal loop and switchabletype-I and type-II weyl fermions in two dimensions, Phys.Rev. B , 075133 (2020).[53] Y.-J. Song and K.-W. Lee, Symmetry-protected spinfulmagnetic Weyl nodal loops and multi-Weyl nodes in 5 d n cubic double perovskites ( n = 1 , ,035155 (2020).[54] R.-W. Zhang, Z. Zhang, C.-C. Liu, and Y. Yao, Nodalline spin-gapless semimetals and high-quality candidatematerials, Phys. Rev. Lett. , 016402 (2020).[55] B. J. Wieder and C. L. Kane, Spin-orbit semimetals inthe layer groups, Phys. Rev. B , 155108 (2016).[56] Z.-M. Yu, W. Wu, Y. X. Zhao, and S. A. Yang, Circum-venting the no-go theorem: A single Weyl point withoutsurface fermi arcs, Phys. Rev. B , 041118 (2019).[57] C. Bradley and A. Cracknell, The Mathematical Theoryof Symmetry in Solids: Representation Theory for PointGroups and Space Groups, The Mathematical Theoryof Symmetry in Solids: Representation Theory for PointGroups and Space Groups (Oxford University Press, NewYork, 1972).[58] J. Zak, Berry’s phase for energy bands in solids, Phys.Rev. Lett. , 2747 (1989). [59] B. I. Halperin, Possible states for a three-dimensionalelectron gas in a strong magnetic field, Jap. J Appl. Phys. , 1913 (1987).[60] F. Tang, Y. Ren, P. Wang, R. Zhong, J. Schneeloch,S. A. Yang, K. Yang, P. A. Lee, G. Gu, Z. Qiao, andL. Zhang, Three-dimensional quantum hall effect andmetal–insulator transition in ZrTe , Nature , 537(2019).[61] B. Lv, T. Qian, and H. Ding, Angle-resolved photoemis-sion spectroscopy and its application to topological ma- terials, Nat. Rev. Phys. , 609 (2019).[62] H. Zheng, S.-Y. Xu, G. Bian, C. Guo, G. Chang,D. S. Sanchez, I. Belopolski, C.-C. Lee, S.-M. Huang,X. Zhang, R. Sankar, N. Alidoust, T.-R. Chang, F. Wu,T. Neupert, F. Chou, H.-T. Jeng, N. Yao, A. Bansil,S. Jia, H. Lin, and M. Z. Hasan, Atomic-scale visualiza-tion of quantum interference on a Weyl semimetal surfaceby scanning tunneling microscopy, ACS Nano10